nLab
hyperbolic manifold

Contents

Context

Riemannian geometry

Manifolds and cobordisms

Contents

Idea

A hyperbolic space is the analog of a Euclidean space as one passes from Euclidean geometry to hyperbolic geometry. The generalization of the concept of hyperbolic plane to higher dimension.

A hyperbolic manifold is a Riemannian manifold (X,g)(X,g) of constant sectional curvature 1-1.

Properties

Zeta functions

There are canonical zeta functions associated with suitable (odd-dimensional) hyperbolic manifolds, see at Selberg zeta function and Ruelle zeta function.

Relation to Chern-Simons theory

There is a curious relation of volumes of hyperbolic 3-manifolds to the action functional of Chern-Simons theory/Dijkgraaf-Witten theory.

Let GG be a Lie group and c:BGB 3U(1)\mathbf{c} \colon \mathbf{B}G \to \mathbf{B}^3 U(1) a cocycle in degree-3 (generalized) Lie group cohomology. Write G\flat G for the underlying discrete group and c:BGB 3U(1)\flat \mathbf{c} \colon \mathbf{B} \flat G \to \mathbf{B}^3 \flat U(1) for the induced cocycle in ordinary (discrete) group cohomology, [c]H Grp 3(G disc,U(1) disc)[\flat \mathbf{c}] \in H^3_{Grp}(G_{disc},U(1)_{disc}).

Then for Σ\Sigma a closed manifold of dimension 3, a map (of smooth infinity-groupoids) ΣBG\Sigma \to \mathbf{B}\flat G is a flat GG-principal connection on Σ\Sigma and the composite

[Σ,BG][Σ,c][Σ,B 3U(1)] ΣU(1) [\Sigma, \mathbf{B}\flat G] \stackrel{[\Sigma, \flat \mathbf{c}]}{\to} [\Sigma, \mathbf{B}^3 \flat U(1)] \stackrel{\int_{\Sigma}}{\to} U(1)

is the action functional for GG-Chern-Simons theory on Σ\Sigma restricted to GG-flat connections, or equivalently is the action functional of G\flat G-Dijkgraaf-Witten theory.

Now for G=SL(n,)G = SL(n,\mathbb{C}) the complex special linear group and hence for Chern-Simons theory with complex gauge group, it turns out that the imaginary part of this flat Chern-Simons/Dijkgraaf-Witten invariant of 3-manifolds always has an expression as a combination of volumes of hyperbolic 3-manifolds.

For n=2n = 2 this is well understood conceptually. For n3n \geq 3 it has been checked by the computer algebra? system SnapPy (Zickert 07), but the conceptual reason remains unclear.

The combination CS(ω)+ivolCS(\omega) + i vol is also called the complex volume or Cheeger-Simons class of a hyperbolic 3-manifold (e.g. Neumann 11, section 2.3. Garoufalidis-Thurston-Zickert 11).

(It is maybe noteworthy that the same kind of combination appears as the contribution of membrane instantons.)

Mostow rigidity theorem

The Mostow rigidity theorem states that every hyperbolic manifold of dimension 3\geq 3 and of finite volume is uniquely determined by its fundamental group.

A Riemannian manifold

  • with zero sectional curvature is a Euclidean manifold?;

  • with +1 sectional curvature is an elliptic manifold?

See also

References

See also

The relation of volumes of hyperbolic 3-manifolds to analytically continued Chern-Simons theory/Dijkgraaf-Witten theory is discussed in

Last revised on April 24, 2018 at 10:38:57. See the history of this page for a list of all contributions to it.